Question

The following region of space has constant charge density 𝜌. Assume that the region is infinite in the x-y plane and of height d (see Figure below).

(a) Determine the electric field inside and outside the charged region.

(b) Given that d is negligible, use Coulombs law to find the electric field above and below the z-axis.

Answer 1

$\Phi=\oint E.ds$

$\Phi=E(4\pi r^2)$

$\Phi=\frac{q}{\epsilon_0}$

Both equation are equally

Out side

$E(4\pi r^2)=\frac{q}{\epsilon}$

$E=\frac{q}{4\pi\epsilon r^2}$

$q=\frac{4\pi R^3\rho}{3}$

$E=\frac{\rho R^3}{3\epsilon r^2}$

Inside

We know that

$\Phi=4\pi r^2 E$

$4\pi r^2E=\frac{qr^3}{\epsilon R^3}$

$E=\frac{kqr}{R^3}$

$E=\frac{\rho r}{3\epsilon}$

$E=\frac{kqz}{(z^2+d^2)^\frac{3}{2}}$

$d<<z$

$E=\frac{kq}{z^2}$